Microscopic study of development of quadruple deformation in neutron-rich Cr isotopes

1. Microscopic study of development of quadruple deformation in neutron-rich Cr isotopes Koichi Sato(Kyoto Univ./RIKEN) Nobuo Hinohara(RIKEN) Takashi Nakatsukasa(RIKEN) Masayuki Matsuo(Niigta Univ.) Kenichi Matsuyangi(RIKEN / YITP)

2. development of deformation in Cr isotopes around N～40 Experimental 2+ excitation energies & E(41+)/ E(21+) ratios N=28 A traditional magic number N=32 A new magicity in neutron-rich nuclei Ca: Huck et al., PRC 31, 2226 (1985). Ti : Janssens et al., PLB(2002) Cr : Prisciandaro et al, PLB 510 (2001) 17 N～40 onset of deformation？ Effect of νg9/2 Sudden rise in R4/2 from to In this work, we study development of deformation in Cr isotopes around N～40 with the CHFB+LQRPA method Gade et al., Phys.Rev.C81 (2010) 051304(R),

3. We introduce “Constrained HFB+ Local QRPA method”, a method of determining microscopically 5D quadrupole collective Hamiltonian (Generalized Bohr-Mottelson Hamiltonian) : collective potential vibrational mass rotational MoI LQRPA masses include the contribution from the time-odd component of the mean field “CHFB+ LQRPA” method is based on the Adiabatic SCC method Matsuo, Nakatsukasa, and Matsuyanagi, Prog.Theor. Phys. 103(2000), 959. N. Hinohara, et al, Prog. Theor. Phys. 117(2007) 451. and an approximation of the 2-dimensional ASCC.

4. Constrained HFB + Local QRPA method Constrained HFB (CHFB) equation: Local QRPA (LQRPA) equations for vibration: Local QRPA equations for rotation: 5D Quadrupole Collective Hamiltonian

5. Classical Quadrupole Collective Hamiltonian: Pauli’s prescription General Bohr-Mottelson Hamiltonian (5D quadrupole collective Hamiltonian): Collective Schrodinger equation: Collective wave function:

6. Application to the low-lying states in neutron-rich Cr isotopes Nuclei 58, 60, 62, 64Cr Microscopic Hamiltonian P+QQ model: s. p. energies + pairing(monopole & quadrupole) + p-h quadrupole int. Model space Nsh=3, 4 for neutrons (pf & sdg shells) Nsh=2, 3 for protons (sd & pf shells) Parameters s. p. energies : modified oscillator monopole pairing strength & quadrupole int. strength 64Cr : adjusted by fitting to the pairing gaps and defomations obtained by Skyrme(SkM*) HFB calculation Stoitsov et al.,Comp. Phys. Com. 167 (2005) 43 62,60,58Cr : assumed simple mass number dependence Baranger & Kumar, NPA110 (1968) 490. quadrupole pairing strength : self-consistent value Sakamoto& Kishimoto. PLB245(1990) 321

7. Collective potential 58Cr 60Cr Prolate minima found in all the nuclei 62Cr 64Cr

8. 62Cr LQRPAMoments of Inertia Strong β-γdep. 58Cr 64Cr Local QRPA vibrational masses:

9. LQRPAMoments of Inertia Strong β-γdep. 58Cr Local QRPA vibrational mass:

10. Excitation Energies Exp.: N. Aoi et al., Nucl. Phys. A805 (2008) 400c S. Zhu et al., Phys. Rev. C74 (2006) 064315. 60Cr 62Cr Our result agrees with the experimental data qualitatively.

11. 60Cr Collective wave functions squared for 60Cr

12. 62Cr Collective wave functions squared for 62Cr

13. EXP：Gade et al., Phys.Rev.C81 (2010) 051304(R), S. Zhu et al., Phys. Rev. C74 (2006) 064315. N. Aoi et al., NPA 805(2008) 400c A. Bürger et al., PLB 622 (2005) 29 (en, ep) =(0.5, 1.5)

14. Summary We have developed a method (CHFB+LQRPA method) of determining the five-dimensional collective Hamiltonian microscopically. We applied this method to the low-lying states in Cr isotopes around N~40. Aside from 64Cr, our results are qualitatively in good agreement with experimental data and suggest that the deformation develops from N=36 to N=38. The interplay of the large-amplitude shape fluctuation in the γ direction, the beta vibrational excitation and rotation, plays an important role. Outlook Comparison with the 1D calculation （only the β degree of freedom) Fully self-consistent 2D Adiabatic SCC method